Surface simplification preserving a solid volume

ABSTRACT

Computer systems may be used to generate and display objects represented by triangles defined by coordinates of vertices. The present invention generates coordinates of a simplified vertex based upon coordinates of vertices adjacent to a first vertex and to a second vertex that define an edge of the triangles. First, a set of triangles that are adjacent to the edge is identified, Second, a first volume associated with the set of triangles is calculated. Finally, the coordinates of the simplified vertex are calculated such that a second volume associated with the simplified vertex corresponds to the first volume.

BACKGROUND OF THE INVENTION

1. Technical Field

The invention relates generally to computer graphics systems, and, moreparticularly, to computer graphics systems that provide real-timeinteractive and/or scientific visualization of complex geometric models.

2. Description of the Related Art

Traditional computer graphics systems are used to create a model ofthree-dimensional objects and render the model for display on atwo-dimensional display device such as a cathode ray tube or liquidcrystal display. Typically, the three-dimensional objects of the modelare each represented by a multitude of polygons (or primitives) thatapproximate the shape of the object. The primitives that define anobject are typically defined by coordinates in a local coordinatesystem. For example, one of the primitives that may be used to define anobject is a triangle, which is defined by the coordinates of threevertices in the local coordinate system.

Such polygonal surfaces may be used for generating pictures andanimations, and may also be used in scientific visualization, ingeometric computing as well as in medical imaging. More specifically,polygonal surfaces have been used for measuring volumes, for aligningthree dimensional scans such as Computed Tomography or MagneticResonance scans, or for finite element applications. IBM VisualizationData Explorer is an example of software that generates and visualizespolygonal surfaces.

The rendering of a polygonal surfaces is divided into two stages:geometry processing and rasterization. Geometry processing typicallyincludes a modeling transformation, lighting calculations, a viewingtransformation, a clipping function, and viewport mapping. The modelingtransformation transforms the primitives from the local coordinatesystem to a world coordinate system. The lighting calculations evaluatean illumination model at various locations: once per primitive forconstant shading, once per vertex for Gouraud shading, or once per pixelfor Phong shading. The viewing transformation transforms the primitivesin world coordinates to a 3D screen coordinate system (sometimesreferred to as the normalized projection coordinate system). Theclipping function determines the primitives (or portions of theprimitives) that are within the viewing frustrum. And viewport mappingmaps the coordinates of the clipped primitives to the normalized devicecoordinate system (sometimes referred to as the 2D device coordinatesystem).

Rasterization is the process which converts the description of theclipped primitives generated during geometry processing into pixels fordisplay. A typical primitive, as shown in FIG. 1A, is a triangle T₁.Other area or surface primitives conventionally are converted into oneor more triangles prior to rasterization. Consequently, the conventionalrasterization process need only to handle triangles. The triangle T₁ isrepresented by the (x,y,z) coordinates at each of its vertices. The(x,y) coordinates of a vertex tell its location in the plane of thedisplay. The z coordinate tells how far the vertex is from the selectedview point of the three-dimensional model. Rasterization is typicallydivided into three tasks: scan conversion, shading, and visibilitydetermination.

Scan conversion utilizes the (x,y) coordinates of the vertices of eachtriangle to compute a set of pixels S which is covered by the triangle.

Shading computes the colors of the set of pixels S covered by eachtriangle. There are numerous schemes for computing colors, some of whichinvolve computationally intensive techniques such as texture mapping.Shading typically utilizes the lighting calculations described abovewith respect to geometric processing.

Visibility determination utilizes the z coordinate of each triangle tocompute the set of pixels S_(v) (a subset of S) which are “visible” forthe triangle. The set SV will differ from the set S if any of the pixelsin set S are covered by previously rasterized triangles whose z valuesare closer to the selected view point. Thus, for each triangle in themodel, a pixel is “visible” if it is in the set S_(v) or “hidden” if itis the set S but not in the set S_(v). Moreover, a triangle is “allvisible” if the set S_(v) is identical to set S, “partially hidden” ifthe set SV is not identical to set S and set S_(v) is not empty, or “allhidden” if set S_(v) is empty. For example, FIG. 1B shows two triangles,triangle T1 and T2, wherein triangle T1 is partially hidden by triangleT2. Visibility determination is traditionally accomplished using az-buffer technique. The Z-buffer stores an entry for each pixel thatrepresents the z-value of the visible primitive at the given pixel. Thez-value (Znew) of a set of pixels covered by a given triangle isdetermined through interpolation. For each pixel within the set, Znew atthe given pixel is compared to the z-value (Zold) stored as an entry inthe Z-buffer corresponding to the given pixel, and the entry Zold isupdated with Znew according to results of the compare operation.

Rasterization is completed by writing the colors of the set of visiblepixels S_(v) to an image buffer for display.

As models become more and more complex, traditional rendering techniquesare too computationally intensive for many graphics systems and resultin degraded performance. For example, it is not rare in medical imagingapplications that a surface may contain hundreds of thousands oftriangles. In such a case, traditional rendering techniques are notsufficient using current computer hardware and software. In order tosolve this problem, several simplification techniques have beendeveloped that construct an approximation of the original surface, andthus reduce the level of detail of the graphical representation of thesurface.

An example of such simplification techniques is described Kalvin et al.“Superfaces: Polygonal Mesh Simplification with Bounded Error”, ComputerGraphics and Applications, Vol. 16, No. 3, May 1996, pp. 64-77, Cohen etal., “Simplification Envelopes”, ACM SIGGRAPH 96, August 1996, AddisonWesley, pp. 119-128, and Ronfard et al., “Full-Range Approximation ofTriangulated Polyhedra”, Computer Graphics Forum, Vol. 15, 1996, pp.67-76. All three algorithms do not undo any simplification that waspreviously made.

The simplification technique of Kalvin et al. attempts to merge surfacetriangles to quasi-planar “superfaces”. Each original triangle is mergedwith one superface, possibly limited to one single triangle. Eachsuperface is guaranteed to lie within a given distance to the trianglesthat were merged into the superface. However, because the algorithm doesnot give priorities to triangles before merging them, the algorithm doesnot produce economical simplifications for a given amount of processingtime. In addition, the algorithm does not preserve the volume of asolid. Finally, the algorithm does not attempt to optimize the aspectratios of triangles.

The simplification technique of Cohen et al. begins by constructing anenvelope around the original surface, whose width corresponds to themaximum distance that is allowed between the simplified surface and theoriginal surface. The envelope does not intersect itself. The surfacevertices are then selected as candidates for removal and the hole leftby the vertex removal is triangulated. The simplification algorithm endswhen no more removal is possible. A verification process verifies thateach new triangle is inside the envelope and that it does not intersectother surface triangles. This algorithm is computationally intensive anddoes not preserve the volume of a solid.

The simplification technique of Ronfard et al. places the edges of thesurface in a priority queue, ordered by a key that measures theapproximation error after the edge collapse. Until the key exceeds auser specified value, the edge with the highest priority is collapsed.This algorithm is computationally intensive because each time an edgecollapse is performed, the simplification error must be re-estimated foreach edge that was adjacent to the collapsed edge. Moreover, thetechnique does not guarantee that the distance between any point of theoriginal surface and the simplified surface and the distance between anypoint of the simplified surface and the original surface will be lessthan a user-specified tolerance, but can only guarantee that a factor ofthis tolerance, depending on the surface geometry, will be respected.Also, the technique does not preserve a volume of a solid.

Moreover, none of the prior art techniques report, for each point of thesimplified surface, a useful bound to the error at that point, which isthe distance to the original surface at that point. Instead, the priorart techniques report a global error bound which is not directly relatedto the particular error at that point.

SUMMARY OF THE INVENTION

The above-stated problems and related problems of the prior art aresolved with the principles of the present invention, surfacesimplification preserving a solid volume and respecting distancetolerances. Computer systems may be used to generate and display objectsrepresented by triangles defined by coordinates of vertices. The presentinvention generates coordinates of a simplified vertex based uponcoordinates of vertices adjacent to a first vertex and to a secondvertex that define an edge of the triangles. First, a set of trianglesthat are adjacent to the edge is identified, Second, a first volumeassociated with the set of triangles is calculated. Finally, thecoordinates of the simplified vertex are calculated such that a secondvolume associated with the simplified vertex corresponds to the firstvolume.

In addition, a technique is presented that generates a second objectwhich is a simplified representation of a first object. The techniquebegins by identifying first and second vertices that define an edge. Thecoordinates of a simplified vertex that corresponds to first and secondvertices of the edge is determined. Error values and tolerance valuesare assigned to vertices. First error volumes corresponding to thevertices of a first set of triangles are derived. The first set oftriangles share at least one of the first and second vertices of theedge. The first error volumes are based upon the error values assignedto vertices of the first set of triangles. A second set of trianglesthat share the simplified vertex is identified and partitioned into aset of planar polygons. Second error volumes corresponding to verticesof the set of planar polygons are derived based upon the first errorvolumes. The second error volumes enclose the first error volumes. Thirderror volumes corresponding to vertices of the second set of trianglesare derived. The third error volumes are based upon the first and seconderror volumes. The third volumes enclose both the first error volumesand the second error volumes. A tolerance volume corresponding to thesimplified vertex is derived. Fianllym, the coordinates of thesimplified vertex is stored in memory for subsequent reuse based upon acomparison operation of the third error volume corresponding to thesimplified vertex and the tolerance volume.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a pictorial representation of the rasterization of a triangleT1;

FIG. 1B is a pictorial representation of the rasterization of trianglesT1 and T2, wherein triangle T1 is partially hidden by triangle T2;

FIG. 2 is a functional block diagram of a graphics work station;

FIG. 3 is a functional block diagram of a graphics subsystem of thegraphics work station of FIG. 2;

FIG. 4 is a pictorial illustration of an entry in the list of verticesused by the preferred embodiment of the present invention;

FIG. 5 is a pictorial illustration of an entry in the list of trianglesused by the preferred embodiment of the present invention;

FIG. 6 is a pictorial illustration of an entry in the list of edges usedby the preferred embodiment of the present invention;

FIG. 7 is a flow chart illustrating the simplification technique of thepresent invention;

FIG. 8 is a pictorial illustration of a collapsible edge and theresulting operations;

FIGS. 9(A)-(C) is a flow chart illustrating the rendering of asimplified model according to the present invention;

FIGS. 10(A)-(C) is a pictorial illustration of the data structures usedby the preferred embodiment of the present invention in representing thesimplified model;

FIG. 11 is a flow chart describing the preliminary collapsibility testswhich are preferably used by the simplification technique of the presentinvention;

FIG. 12 is a flow chart illustrating the operations in determining thecoordinates of the potential simplified vertex according to the presentinvention;

FIG. 13(A) is a pictorial representation of a solid;

FIG. 13(B) is a pictorial representation of a star of a vertex;

FIG. 13(C) is a pictorial representation of a link of a vertex;

FIG. 13(D) represents a regular vertex;

FIG. 13(E) represents a boundary vertex;

FIG. 13(F) represents a singular vertex;

FIG. 13(G) is a pictorial representation of an edge star beforesimplification;

FIG. 14 is a pictorial representation of the star of a potentialsimplified vertex;

FIG. 15 is a flow chart illustrating the secondary collapsibility testswhich are preferably used by the simplification technique of the presentinvention;

FIG. 16 is a flow chart illustrating operation of the error tolerancetest of FIG. 15 according to the present invention;

FIG. 17 is a pictorial representation of the edge star of FIG. 13(G)illustrating the assignment of error values and tolerance values to thevertices of the edge star prior to simplification.

FIG. 18 is a pictorial representation of star of FIG. 14 illustratingthe assignment of error values and tolerance values to the vertices ofthe star of the potential simplified vertex that results fromsimplification.

FIG. 19 is a pictorial representation of an exemplary case of thedetermination of a crossing that occurs in a Type A constraint;

FIG. 20 is a pictorial representation of an exemplary case of thedetermination of barycentric coordinates of a point with respect to agiven triangle;

FIG. 21 is a pictorial representation of the edge star and associatedparameters used in the determination of Type B constraints;

FIG. 22 is a pictorial representation of the edge star and associatedparameters used in the determination of Type C constraints;

FIG. 23 is a pictorial representation of an exemplary case of thedetermination of whether a given vertex is located within a giventriangle;

FIGS. 24(A)-(B) are pictorial representations of the edge star andsimplified vertex star, respectively, illustrating notations used inMethods IV and V of the present invention; and

FIG. 25 illustrates the processing of a boundary edge according to analternate embodiment of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

As shown in FIG. 2, a conventional graphics system 100 includes a hostprocessor 102 which is coupled to a system memory 104 via a system bus106. The system memory 104 consists of random access memory (RAM) thatstores graphics data defining the objects contained in one or more threedimensional models. The graphics data that defines each object consistsof coordinates in a local coordinate system and attributes (e.g. color,reflectance, texture) of primitives. The primitives are geometricentities such as a polygon, line or surface. Typically, the primitivesare triangles defined by the coordinates of three vertices in the localcoordinate system. In the event that a model includes non-triangularprimitives, such primitives may be approximated by planar polygons(i.e., tesselated) and such planar polygons may be triangulated.Triangulation may be accomplished using various known algorithms andneed not introduce any new vertices into the model. As a result of thetriangulation operation, each face (or surface) of the original model isdecomposed into one or more non-overlapping triangular primitives.

The system memory 104 includes an ordered list of vertices of thetriangles that define the surfaces of objects that make up a threedimensional model. In addition, the system memory 104 may store a listof triangle identifiers that correspond to each of the triangles andtransformation matrices. The transformation matrices are used totransform the objects of the model from the local coordinate system to aworld coordinate system, and thus specify the position, orientation andscale of the triangles in the model.

Input/output (I/O) devices 108 interface to the host processor 102 viathe system bus 106. The I/O devices may include a keyboard, template ortouch pad for text entry, a pointing device such as a mouse, trackball,Spaceball or light pen for user input, and non-volatile storage such asa hard disk or CD-ROM for storing the graphics data and any applicationsoftware. As is conventional, the graphics data and application softwareis loaded from the non-volatile storage to the system memory 104 foraccess by the system processor 102.

The application software typically includes one or more user interfacesthat provides the user with the ability to update the view point (orcamera) and thus navigate through the model. In addition, theapplication software typically includes one or more user interfaces thatprovide the user with the ability to perform animations, which is theview from a series of pre-defined view points. When the view point isupdated, the application software typically utilizes the transformationmatrices stored in the system memory 104 to transform the primitives ofthe model from the local coordinate system to a world coordinate system.The application software executing on the host processor 102 thensupplies a graphics order to render the model to a graphics subsystem110 that interfaces to the system memory 104 via the system bus 106.

Generally, the graphics subsystem 110 operates to render the graphicsdata stored in the system memory 104 for display on a display area of adisplay device 112 according to the graphics orders transferred from thehost processor 102 to the graphics subsystem 110. The display device 112may utilize raster scan techniques or liquid crystal display techniquesto display the pixels. The pixel data generated by the graphicssubsystem 110 is in digital form. Typically, the display device 112requires the pixel data in analog form. In this case, as shown in FIG.3, a digital-to-analog converter 114 may be placed between the graphicssubsystem 110 and the display device 112 to convert the pixel data froma digital to an analog form.

The graphics orders typically consist of a sequence of data blocks thatinclude, or point to, the graphics data (e.g. coordinates and attributesof one or more primitives) that defines the primitives of the model,associated transformation matrices, and any other necessary informationrequired by the graphics subsystem 110. The primitives associated withthe graphics orders are typically defined by the value of the geometriccoordinates or homogeneous coordinates for each vertex of the primitive.In addition, graphics orders typically include, or point to, datadefining normal vectors for the vertices of each primitive. The valuesof these coordinates and normal vectors are typically specified in theworld coordinate system.

In addition, the transformation of the primitives of the model from thelocal coordinate system to a world coordinate system may not beperformed by the application executing on the host processor, but may beperformed by the graphics subsystem. In this case, the graphics ordersupplied to the graphics subsystem includes, or points to thetransformation matrices stored in the system memory 104, and thegraphics subsystem utilizes the transformation matrices to transform theprimitives of the model from the local coordinate system to a worldcoordinate system.

Although the graphics subsystem 110 is illustrated as part of a graphicswork station, the scope of the present invention is not limited thereto.Moreover, the graphics subsystem 110 of the present invention asdescribed below may be implemented in hardware such as a gate array or achip set that includes at least one programmable sequencer, memory, atleast one integer processing unit and at least one floating pointprocessing unit, if needed. In addition, the graphics subsystem 110 mayinclude a parallel and/or pipelined architecture as shown in U.S. Pat.No. 4,876,644, commonly assigned to the assignee of the presentinvention and incorporated by reference herein in its entirety.

In the alternative, the graphics subsystem 110 (or portions thereof) asdescribed below may be implemented in software together with aprocessor. The processor may be a conventional general purposeprocessor, the host processor 128, or a co-processor integrated with thehost processor 128.

As shown in FIG. 3, the graphics subsystem 110 includes a control unit200 that supervises the operation of the graphics subsystem 110. Uponreceiving a graphics order to render a model, the control unit 200passes the graphics data associated with the graphics order on to ageometry engine 202. The geometry engine 202 transforms the graphicsdata associated with the graphics order from the world coordinate systemto a 3D screen coordinate system (sometimes referred to as a normalizedprojection coordinate system), clips the graphics data against apredetermined view volume, and transforms the clipped data from the 3Dscreen coordinate system to a normalized device coordinate system(sometimes referred to as the 2D device coordinate system or the screencoordinate system). In addition, depending upon the shading algorithm tobe applied, an illumination model is evaluated at various locations(i.e., the vertices of the primitives and/or the pixels covered by agiven primitive). The transformed and clipped graphics data defined bycoordinates in the normalized device coordinate system is then passed onto a rasterization stage 212.

The rasterization stage performs scan conversion thereby converting thetransformed primitives into pixels, and performs shading calculationsand visibility determination which generally stores each primitive'scontribution at each pixel in at least one frame buffer 216 and az-buffer 214. The Z-buffer 214 preferably contains sufficient memory tostore a depth value for each pixel of the display 112. Conventionally,the depth value is stored as a 24-bit integer for each pixel. The framebuffer 216 preferably contains sufficient memory to store color data foreach pixel of the display 112. Conventionally, the color data consistsof three 8-bit integers representing red, green and blue (r,g,b) colorvalues for each pixel. The pixel data is periodically output from theframe buffer 216 for display on the display device 112. Thefunctionality of the geometry engine 202 and rasterization stage 212 maybe organized in a variety of architectures. A more detailed discussionof such architectures may be found in Foley et. al., “Computer Graphics:Principles and Practice”, pp. 855-920 (2nd Ed. 1990), hereinincorporated by reference in its entirety.

According to the present invention, a mechanism is provided thatconstructs a simplified approximation of one or more objects of a modelto produce a simplified model. The mechanism may be executed on anoriginal non-simplified model. In the alternative, the mechanism may beexecuted on a simplified model and thus employ the result of a previoussimplification as the starting point for subsequent simplification.Preferably, the mechanism of the present invention is integrated intothe application software that is loaded from non-volatile storage to thesystem memory 104 where it is executed by the system processor 102. Inaddition, the mechanism may be contained in a graphics library that ispart of an application developers kit.

Generally, the simplification mechanism of the present invention is bestsuited for producing simplifications of solid objects that enclose theexact same volume as the original solid object. An example of a solid isshown in FIG. 13(A). A definition of a solid can be found in C.Hoffmann's “Geometric and Solid Modeling”, Morgan Kaufmann, 1989, pp51-65, herein incorporated by reference in its entirety. A solid may bepartitioned into two classes: “solid without boundary” and “solid withboundary”. A “solid without boundary” is made exclusively of regularvertices as set forth below. A “solid with boundary” is made of regularvertices and at least three boundary vertices. An example of a regularvertex is shown in FIG. 13(D). An example of a boundary vertex is shownin FIG. 13(E). The simplification mechanism may be utilized with bothclasses of solids.

Moreover, the simplification mechanism of the present invention is bestsuited for triangles that satisfy the following properties: each pair oftriangles is either non-intersecting or intersects exactly at a vertexof each of them or at an edge of each of them. At each vertex, the linkmust be a simple polygonal curve ( the set of triangles that share thatvertex can be ordered). If furthermore, the link is a simple closedpolygonal curve, then the vertex is said to be a “regular vertex”. Anexample of a link of a vertex is shown in FIG. 13(C). If the link is asimple open polygonal curve, then the vertex is said to be on theboundary, or a “boundary vertex”. Any vertex that is neither regular norboundary is a “singular vertex”. An example of a singular vertex isshown in FIG. 13(F). A more detailed description of a link may be foundat C. Hoffmann's “Geometric and Solid Modeling”, Morgan Kaufmann, 1989,pp 53-54. However, it should be understood that such constraints placedupon on the modeling geometry is not to be construed in a limiting senseupon the practice of the present invention.

The simplification mechanism of the present invention preferably storesin system memory 104 a list of vertices 300, a list of triangles 400 anda list of edges 500. It should be realized that these particular datastructures are not to be construed in a limiting sense upon the practiceof the present invention.

The list of vertices 300 include an entry associated with each vertex ofthe triangles that represent the object(s) of a model. As shown in FIG.4, each entry preferably includes a true ID field 302, a representativeID field 304, coordinate fields 306 associated with a given vertex. Inaddition, the entry may include an error field 308 and a tolerance field310 associated with the given vertex. The functionality of the errorfield 308 and tolerance field 310 is discussed below with respect toFIG. 16. The coordinate fields 306 store the coordinates (such as thex,y,z coordinates) of the given vertex. The true ID field 302 stores anidentifier assigned to the given vertex. And the representative ID field304 stores an identifier that points to a vertex that represents thegiven vertex. Preferably, the representative ID field 304 is initializedto point to the given vertex and thus initially stores the same value asthe true ID field 304. However, as described in more detail below, whenthe simplification mechanism of the present invention collapses an edge,a simplified vertex replaces the vertices of the edge. Preferably, thesimplified vertex is added to the list of vertices by:

a) updating the coordinates fields 306 for one vertex of the collapsededge; and

b) updating the representative ID field 304 for the other vertex of thecollapsed edge to point to the other/moved vertex of the collapsed edge.It should be noted that the entries of the list of vertices may excludethe true ID field 302 if the identifier assigned to the givenvertex/entry corresponds to the location of the entry in the list ofvertices. In other words, the true ID field 302 is implicitly assignedto the entry in this case.

The list of triangles 400 include an entry associated with each trianglethat represents the object(s) of a model. As shown in FIG. 5, each entrypreferably includes a true ID field 402, a representative ID field 404,vertex identifier fields 406, and normal fields 408 that are associatedwith a given triangle. The normal fields 408 store data that representscomponents of the surface normal (such as the nx,ny,nz components) atthe given triangle. Such a surface normal is typically used duringrasterization for lighting calculations. The vertex identifier fields406 preferably identify the vertices vi,vj,vk of the associated triangleprior to simplification. Thus, the vertex identifier fields 406preferably store the value of the true ID field 302 for the verticesvi,vj,vk. Preferably, the vertex identifier fields 406 for the verticesvi,vj,vk are ordered by convention, for example, by a convention whereinthe vertices vi,vj,vk are oriented sequentially in a counter-clockwisefashion. The true ID field 402 stores an identifier assigned to thegiven triangle. And the representative ID field 404 stores an identifierthat points to a triangle that represents the given triangle.Preferably, the representative ID field 404 is initialized to point tothe given triangle and thus initially stores the same value as the trueID field 404. However, as described in more detail below, when thesimplification mechanism of the present invention collapses an edge, thetwo triangles that share the collapsed edge are replaced byrepresentative triangles. Preferably, the two triangles that share thecollapsed edge are replaced by representative triangles by updating therepresentative ID field 404 for the replaced triangles. In addition, foreach triangle that shares one of the vertices of the collapsed edge(except for the two replaced triangles that share the collapsed edge),new surface normals are computed for the given triangle and the normaldata 408 associated with the given triangle is updated. It should benoted that the entries of the list of triangles may exclude the true IDfield 402 if the identifier assigned to the given triangle/entrycorresponds to the location of the entry in the list of triangles. Inother words, the true ID field 402 is implicitly assigned to the entryin this case.

The list of edges 500 include an entry associated with each edge thatrepresents the triangles of the object(s) of a model. As shown in FIG.6, each entry preferably includes a true ID field 502, a representativeID field 504, vertex identifier fields 506, and triangle identifierfields 508 that are associated with a given edge. The vertex identifierfields 506 preferably identify the vertices v1,vm of the given edgeprior to simplification. Thus, the vertex identifier fields 506preferably store the value of the true ID field 302 for the verticesvl,vm. Preferably, the vertex identifier fields 506 for the verticesv1,vm of the given edge are ordered by convention, for example, by theconvention where the value stored by the true ID field 302 for vertex v1is greater than the value stored by the true ID field 302 for the vertexvm. The triangle identifier fields 508 preferably identify the trianglesTa,Tb that share the given edge prior to simplification. Thus, thetriangle identifier fields 508 preferably store the value of the true IDfield 402 for the triangles Ta,Tb. Preferably, the triangle identifierfields 508 are ordered by convention, for example, by a conventionwherein the triangles Ta and Th are oriented sequentially in a clockwisesense with respect to the vertex v1 identified by the vertex identifierfields 506. The true ID field 502 stores an identifier assigned to thegiven edge. And the representative ID field 504 stores an identifierthat points to an edge that represents the given edge. Preferably, therepresentative ID field 504 is initialized to point to the given edgeand thus initially stores the same value as the true ID field 504.However, as described in more detail below, when the simplificationmechanism of the present invention collapses an edge, edges thatneighbor the collapsed edge may be replaced by representative edges.Preferably, the neighbor edges are replaced by representative edges byupdating the representative ID field 504 for the replaced edges. Itshould be noted that the entries of the list of edges may exclude thetrue ID field 502 if the identifier assigned to the given edge/entrycorresponds to the location of the entry in the list of edges. In otherwords, the true ID field 502 is implicitly assigned to the entry in thiscase.

In addition, the simplification mechanism of the present inventionpreferably stores in system memory 104 a priority queue 600 whichrepresents a list of edges ordered by priority. The priority queueallows the simplification mechanism of the present invention to make onepass through the edges of the model when constructing the simplifiedmodel. The priority of a given edge is determined by the value of a keyassociated with the edge. The value of the key associated with a givenedge is preferably based upon the length of the given edge plus the sumof the error values associated with the vertices of the given edge. Amore detailed description of such error values is set forth below. And,a more detailed description of the priority queue 600 may be found inCormen et al., “Introduction to Algorithms”, McGraw Hill, 1994, pp.149-151, herein incorporated by reference in its entirety.

FIGS. 7(A)-(B) is a flow chart that illustrates the simplificationtechniques employed by the present invention. In step 601, the edge withthe highest priority is selected from the priority queue 600. In step603, a copy is made of entries in the list of triangles 400 associatedwith the triangles of an edge star that corresponds to the edge selectedin step 601. A more detailed description of an edge star is set forthbelow with respect to step 1105 of FIG. 11. In addition, a copy is madeof the entries of the list of vertices 300 and list of edges 500 thatbelong to one or more triangles of the edge star.

In step 605, one or more preliminary collapsibility tests are performedwith respect to the selected edge. A more detailed description of thepreliminary collapsibility test(s) for the selected edge is describebelow with respect to FIG. 11. The preliminary collapsibility test(s)determines whether or not the edge is collapsible, and appropriatelymarks a flag indicating the collapsibility of the edge. In step 607, itis determined if the flag indicates that the edge is collapsible. If theedge is collapsible, operation continues to step 609; however, if theedge is not collapsible (i.e., failed the preliminary collapsibilitytest(s)), operation continues to step 625.

In step 609, the coordinates of a potential simplified vertexcorresponding to the selected edge is identified and operation continuesto steps 611-613. A more detailed description of the preferred operationof the system in determining the coordinates of the potential simplifiedvertex is set forth below with respect to FIG. 12. Steps 611-613 aredescribed below with respect to FIG. 8 which illustrates an example of acollapsible edge and the resulting operations. In FIG. 8, triangles t0and t1 share edge e0, which has been selected as the collapsible edge.

In step 611, a vertex update operation is performed on the copy of theentries of the list of vertices generated in step 603. Morespecifically, one vertex v0 of the selected edge e0 is selected and thecoordinate fields 306 of the copy of the entry in the list of vertices300 that corresponds to the selected vertex v0 is updated to correspondto the coordinates of the simplified vertex (not shown). In addition,the representative ID field 304 of the copy of the entry in the list ofvertices 300 that corresponds to the other vertex v1 of the selectededge e0 is updated to point to the selected vertex v0.

In step 613, an edge update operation is performed on the copy of theentries of the list of edges generated in step 603 wherein two edgesthat neighbor the selected edge are replaced by representative edges.Preferably, the replaced neighboring edges are those edges that sharethe replaced vertex and are part of the replaced triangles. For example,edges e3 and e4 (which share the replaced vertex v1 and are part of thereplaced triangles to,t1) may be replaced by representative edges e1 ande2, respectively. In this example, the neighbor edges e3 and e4 arepreferably replaced by the representative edges e1 and e2 by updatingthe representative ID field 504 of the copy of the entries of the listof edges that correspond to the replaced edges e3 and e4 to point to thetrue ID fields 502 associated with the edges e1 and e2, respectively.

In step 615, a triangle update operation on the copy of entries of thelist of triangles generated in step 603 wherein two triangles t0 and t1that share the collapsed edge e0 are replaced by representativetriangles. Preferably, the representative triangles are those trianglesthat include the edges replaced in step 607. For example, triangles t0and t1 may be replaced by representative triangles t5 and t7,respectively. In this example, the triangles t0 and t1 are replaced bythe representative triangles t5 and t7 by updating the representative IDfield 404 of the copy of the entries of the list of triangles thatcorrespond to the replaced triangles t0 and t1 to point to the true IDfields 402 associated with the triangles t5 and t7, respectively. Inaddition, for each triangle that shares one of the vertices of theselected edge e0 (except for the two replaced triangles that share theselected edge e0), new surface normals are computed for the giventriangle and the normal data 408 associated with the copy of the entryin the list of triangles that corresponds to the given triangle isupdated. In the example shown, new surface normals would be computed forthe triangles t2,t3,t4,t5,t6,t7, and would not be computed for thetriangles t0,t1.

In step 617, one or more secondary collapsibility tests are performedwith respect to the selected edge. A more detailed description of thesecondary collapsibility test(s) for the selected edge is describedbelow with respect to FIG. 15. The secondary collapsibility test(s)determines whether or not the edge is collapsible, and appropriatelymarks the flag indicating the collapsibility of the edge. In step 619,it is determined if the flag indicates that the edge is collapsible. Ifthe edge is collapsible, operation continues to step 621; however, ifthe edge is not collapsible (i.e., failed the secondary collapsibilitytest(s), operation continues to step 625.

In step 621, the list of vertices 300, list of triangles 400 and list ofedges 500 are updated with the corresponding copy of such entries. Thisoperation updates such lists to reflect the simplification that resultsfrom collapsing the selected edge.

In step 623, a priority queue update operation is performed wherebyentries that correspond to the edges replaced in step 613 are removedfrom the queue 600. In addition, those entries of the priority queue 600that correspond to edges having as a vertex one of the vertices of thecollapsed edge (in this example edges e1 and e2) are selected and thepriority for the selected entry is updated. After step 623, operationcontinues to step 627.

In step 625, the selected edge is removed from the priority queue 600and the processing continues to 627.

In step 627, it is determined if the last edge in the priority queue 600has been processed. If not, operation returns to step 601 to process thehighest priority edge in the priority queue 600. If so, thesimplification operation ends.

FIGS. 9(A)-(C) illustrates the operation of the graphics system inrendering the simplified model generated with the simplificationtechniques set forth above with respect to FIG. 7. The renderingoperation first loops through the list of vertices 300 to build a listof simplified vertices 300 a and a correspondence table 900. The list ofsimplified vertices 300 a excludes entries associated with verticesremoved in step 605 above. The operation then loops through the list oftriangles 400 to build a list of simplified triangles 400 a, whichexcludes entries associated with triangles removed in step 609 above.The generation of the list of simplified triangles 400 a utilizes thelist of simplified vertices 300 a and the correspondence table 900. Thelist of simplified triangles 400 a is then forwarded to the graphicssubsystem 110 wherein the simplified triangles are rendered for display.

More specifically, the rendering operation begins in step 901 byinitializing a counter, for example to a zero value. The operation thenperforms a loop 903-911 through the entries in the list of vertices 300.For each entry in the list of vertices 300, the operation performs steps905-909. In step 905, it is determined if the true ID field 302 of entryis equal to the representative ID field 304 of the entry. This stepidentifies whether or not the particular entry is associated with avertex removed in step 611 above. If in step 905 it is determined thatthe true ID field 302 of entry is equal to the representative ID field304 of the entry, operation continues to step 907; otherwise operationcontinues to the next entry in the list of vertices 300. In step 907, anentry is added to the list of simplified vertices 300 a at a locationcorresponding to the current counter, and an entry is added to thecorrespondence table 900. Finally, in step 909 the counter isincremented and operation continues to the next entry in the loop903-911.

As shown in FIG. 10(A), the entry in the list of simplified vertices 300a preferably includes new true ID field 1002 and coordinate fields 1006associated with the given vertex. The new true ID field 1002 stores anew identifier assigned to the vertex after simplification. The newidentifier is preferably set to the current value of the counter. Thecoordinate fields 1006 store the coordinates (such as the x,y,zcoordinates) of the given vertex, and thus store the same data as thecorresponding entry in the list of vertices 300. It should be noted thatthe entries of the list of simplified vertices may exclude the new trueID field 1002 if the identifier assigned to the given vertex/entrycorresponds to the location of the entry in the list of simplifiedvertices 1002. In other words, the new true ID field 1002 is implicitlyassigned to the entry in this case.

As shown in FIG. 10(B), the entry in the correspondence table 900preferable includes an old true ID field 950 and new true ID field 952which encode the correspondence between the identifiers assigned to thegiven vertex before and after simplification. More specifically, the oldtrue ID field 950 stores the value of the identifier stored in the trueID field 302 in the corresponding entry in the list of vertices 300. Thenew true ID field 952 stores the new identifier assigned to the givenvertex, and thus stores the same data as the new true ID field 1002 ofthe corresponding entry in the list of simplified vertices 300 a asshown in FIG. 10(A). It should be noted that the entries of thecorrespondence table 900 may exclude the old true ID field 950 if theold identifier assigned to the given vertex/entry corresponds to thelocation of the entry in the correspondence table 900. In other words,the old true ID field 950 is implicitly assigned to the entry in thiscase.

After completing the loop 903-911, operation continues to step 913wherein a counter is initialized, for example to a zero value. Theoperation then performs a loop 915-933 through the entries in the listof triangles 400. For each entry in the list of triangles 400, theoperation performs steps 917-931. In step 917, it is determined if thetrue ID field 402 of entry is equal to the representative ID field 404of the entry. This step identifies whether or not the particular entryis associated with a triangle removed in step 615 above. If in step 917it is determined that the true ID field 402 of entry is equal to therepresentative ID field 404 of the entry, operation continues to step919; otherwise operation continues to the next entry in the list oftriangles 400. In step 919, a loop 919-927 is performed for the vertexID fields 406 of the entry. For each vertex ID, the operation performssteps 921-925. In step 921, the vertex ID is read from the appropriatevertex ID field 406. As described above with respect to FIG. 4, thevertex ID field points to the true ID field 302 of a vertex/entry in thelist of vertices 300. In step 922, the representative identifier field304 is read from the entry in the list of vertices 300 whose trueidentifier field 302 corresponds to the value stored by vertex ID fieldread in step 921. In step 923, the entry of the correspondence table 900whose old true ID field 950 matches the representative identifier field304 read in step 922 is identified. Finally, in step 925 the valuestored in the new true ID field 952 for the entry of the correspondencetable 900 identified in step 923 is stored. After looping through thevertex ID fields 406 of the entry in the list of triangles 400, in step929 an entry is added to the list of simplified triangles 400 a at alocation corresponding to the current counter.

As shown in FIG. 10(C), the entry in the list of simplified triangles400 a preferably includes new true ID field 1012, vertex identifierfields 1014, and normal fields 1016 associated with the given triangle.The true ID field 1012 stores a new identifier assigned to the giventriangle. The new identifier is preferably set to the current value ofthe counter. The vertex identifier fields 1016 identify the vertices ofthe associated triangle after simplification. Thus, the vertexidentifier fields 1016 preferably store the new true ID values stored instep 925. Finally, the normal fields 1018 store data that representscomponents of the surface normal (such as the nx,ny,nz components) atthe given triangle, and thus store the same data as the correspondingentry in the list of triangles 400. It should be noted that the entriesof the list of simplified triangles 400 a may exclude the true ID field1012 if the identifier assigned to the given triangle/entry correspondsto the location of the entry in the list of simplified triangles. Inother words, the true ID field 1012 is implicitly assigned to the entryin this case.

In step 931, the counter is incremented and operation continues to thenext entry in the loop 915-933.

After completing the loop 915-933, operation continues to step 935. Instep 935, the list of simplified triangles 400 a is communicated to thegraphics subsystem wherein the simplified triangles are rendered fordisplay.

A more detailed description of the preliminary collapsibility test(s) ofstep 605 outlined above is now set forth with reference to FIG. 11. Asdescribed below, such preliminary tests may include one of more of aboundary test, a valence test and a manifold test. Preferably, thepreliminary collapsibility test operation begins in step 1101 byperforming a boundary test with respect to the selected edge. Theboundary test determines if any of the vertices of the selected edge isa boundary vertex. A detailed description of a boundary vertex is setforth above. If in step 1101 it is determined that one or more of thevertices of the selected edge is a boundary vertex, then the selectededge is marked as not collapsible and the operation ends; otherwise theoperation continues to step 1103.

In step 1103, a valence test may be performed with respect to theselected edge. The valence of a vertex is the number of triangles thatshare that vertex. The valence of an edge is represented by the sum ofthe valence of the vertices of the edge. Preferably, the valence test ofstep 1103 determines whether the valence of the edge falls within apredetermined bound. For example, the valence test may be represented asfollows:

3≦(v(v ₁)+v(v ₂)−4)≦valence_(max)

where v(v₁) and v(v₂) represents the valence of two vertices of theedge, and valence_(max) is a predetermined integer value that may bespecified by the user.

If in step 1103 it is determined that the valence associated withselected edge falls outside the predetermined bounds, the selected edgeis marked as not collapsible and the operation ends; otherwise theoperation continues to step 1105.

In step 1105, the edge star of the selected edge is constructed. Theedge star represents a union of the stars of the vertices of theselected edge. A more detailed description of the star of a vertex isdescribed in R. Ronfard and R. Rossignac, “Full-Range Approximation ofTriangulated Polyhedra”, Computer Graphics Forum, Vol. 15, 1996, pp.67-76, herein incorporated by reference in its entirety. An example of astar is shown in FIG. 13(B).

In step 1107, a manifold test may performed to ensure that the verticesof the link of the edge star constructed in step 1105 are different. Thelink of an edge star is equivalent to the link of a vertex formed bycollapsing the edge. A detailed description of the link of a vertex isset forth above. If the manifold test fails, the edge is marked as notcollapsible and the operation ends; otherwise the operation ends.

A more detailed description of the preferred steps in determining thecoordinates of the potential simplified vertex is now set forth withreference to FIG. 12. These steps advantageously place the simplifiedvertex in a specific location such that a volume associated with theedge prior to simplification (i.e. prior to edge collapse) will remainunaltered after simplification (i.e., after edge collapse). Morespecifically, the operation begins in step 1201 by defining a volume vassociated with the selected edge. Preferably the volume v is determinedas follows. First, the coordinates gx,gy,gx of a point in the vicinityof the set of vertices in the edge star of the selected edge.Preferably, the point gx,gy,gz is the centroid of the set of vertices ofthe edge star. Alternatively, the point gx,gy,gz may be assigned to themidpoint between the vertices of the collapsed edge or other similarcoordinates. Second, the tetrahedra s₀,s₁,s₂ . . . s_(m) are identified,where (m+1) is the number of triangles in the edge star of the selectededge, and wherein the vertices of a given tetrahedron s_(i) include thethree vertices of the corresponding triangle t_(i) and the pointgx,gy,gz. For example, as shown in FIG. 13(G), the four vertices of thetetrahedron s1 (cross-hatched) include the three vertices v0,v1,v2 aswell as the point g. Finally, the volume v is calculated as the sum ofthe volumes of the tetrahedra s₀,s₁,s₂ . . . s_(m). The volume of agiven tetrahedron s_(i), which is defined by the vertices (g,vi,vj,vk),is preferably determined by applying a translation T to the vertices(vi,vj,vk) and by computing (⅙) of the determinant of the followingmatrix m: $m\quad = \quad \begin{bmatrix}{vix} & {vjx} & {vkx} \\{viy} & {vjy} & {vky} \\{viz} & {vjz} & {vkz}\end{bmatrix}$

wherein vix,viy, viz denote the translated x y z coordinates of thevertex vi,

wherein vjx, vjy, vjz denote the translated x y z coordinates of thevertex vj, and

wherein vkx, vky, vkz denote the translated x y z coordinates of thevertex vk.

Preferably, the parameters Tx,Ty,Tz of the translation T are set to(−gx,−gy,−gz) of the point g. The determinant of m is preferablycalculated by first performing a QR Decomposition of the matrix m (SeeGolub and Van Load, “Matrix Computations”, Johns Hopkins Press, 1989, pp211ff), which results in the decomposition m=qr, and then by multiplyingtogether the diagonal elements of r.

In step 1203, the potential simplified vertex is assumed to have thecoordinates x,y,z, and an equation is generated that represents a volumeassociated with the potential simplified vertex. Preferably, theequation is generated as follows. First, the tetrahedra r₂,r₃, . . .r_(m-1), r_(m) are identified as follows:

r₂=(g,vs,v2,v3)

r₃=(g,vs,v3,v4)

r₄=(g,vs,v4,v5)

r₅=(g,vs,v5,v6) . . .

r_(m-1)=(g,vs,vm−1,vm)

r_(m)=(g,vs,vm,v2)

An example of such tetrahedra is shown in FIG. 14. Second, an equationfor the volume of each of the tetrahedron r₂,r₃, . . . r_(m) isgenerated. Preferably, the equation of the volume of a given tetrahedronr_(i), which is defined by the vertices (g,vi,vj,vk), is derived as (⅙)of the determinant of the matrix m described above. For example, theequation of the volume for the r₅ tetrahedron may be derived as (⅙) thedeterminant of the matrix m4: ${m5} = \begin{bmatrix}x & {v5x} & {v6x} \\y & {v5y} & {v6y} \\z & {v5z} & {v6z}\end{bmatrix}$

This would lead to the following equation:

({fraction (1/6)})[(x(v5y v6−v5z v6y)+y(v6x v5z−v6z v5x)+z(v5x v6y−v5yv6x)]

Similar formulas hold true for each of the tetrahedra r₂,r₃, . . .r_(m). Finally, the x, y and z components of the equations for eachtetrahedron r₂,r₃, . . . r_(m) are added together to form an equationthat represents a volume associated with the potential representativevertex. The equation is preferably in the form:

x e₁+y e₂+z e₃.

where e₁ represents all of the coefficients of x in the expression forall of the tetrahedra r₂,r₃, . . . r_(m), e₂ represents all of thecoefficients of y in the expression for all of the tetrahedra r₂,r₃, . .. r_(m), and e₃ represents all of the coefficients of z in theexpression for all of the tetrahedra r₂,r₃, . . . r_(m).

In step 1205, the equation generated in step 1203 is equated to thevolume v calculated in step 1201, thus resulting in the followingequation:

x e₁ +y e₂ +z e₃ =v

In step 1207, a plane p_(s) that satisfies the equation generated instep 1205 is identified. Preferably, the plane p_(s) is computed asfollows. The left and right side of the equation generated in step 1205is divided by e₁ ²+e₂ ²+e₃ ² to obtain the following equation, whichdefines the plane p_(s):

x f₁ +y f₂ +z f₃ =v ₁

where f₁ ²+f₂ ²+f₃ ²=1

In step 1209, the equations of the planes p₀, p₁ . . . p_(m) arecomputed, wherein p_(i) is the plane on which the corresponding triangleT_(i) lies. Preferably, the equations of the planes p₀, p₁ . . . p_(m)are computed using the surface normal of the corresponding triangles.More specifically, the equation of a plane p_(i) may be represented asfollows:

p_(i)=a_(i) x+b_(i) y+c_(i) z−d_(i).

The nx,ny, and nz components of the surface normal may be used as thea_(i), b_(i), c_(i) coefficients of the plane equation. The d_(i)coefficient may be obtained by computing the scalar product between thesurface normal and each vertex of the corresponding triangle, andaveraging such scalar products.

Finally, in step 1211, the coordinates x,y,and z of the potentialsimplified vertex vs are solved such that x,y,z of vs lie in the planep_(s) and the sum of the squared distances from vs to each of the planesp₀, p₁ . . . p_(m) is a minimum. This is preferably accomplished asfollows. First, a symmetry transformation S is determined that exchanges(f₁, f₂, f₃) of the plane equation generated in step 1207 with (0, 0,1). The symmetry transformation S is preferably determined using aHouseholder transformation (See Golub and Van Loan, pp 195-197). Second,the transformation S is applied to the coordinates of each vertex v0,v1,. . . , vm of the triangles of edge star as well as the equation of eachplane p₀, p₁ . . . , p_(m) generated in step 1209. Third, a linearsystem of equations is computed as follows: ${\begin{bmatrix}a_{0} & b_{0} \\a_{1} & b_{1} \\a_{2} & b_{2} \\... & \quad \\a_{m} & b_{m}\end{bmatrix}\quad\begin{bmatrix}x \\y\end{bmatrix}}\quad = \quad \begin{bmatrix}{{c_{0}v_{1}} - \quad d_{0}} \\{{c_{1}v_{1}} - \quad d_{1}} \\{{c_{2}v_{1}} - \quad d_{2}} \\... \\{{c_{m}v_{1}} - \quad d_{m}}\end{bmatrix}$

Fourth, the set of linear equations is solved for x and y using standardNormal Equations, for example, as described in Golub and Van Loan,“Matrix Computations”, Johns Hopkins Press, 1989, pp. 224-225,incorporated herein by reference in its entirety. In addition, the valueof z is set to v₁. Finally, the transformation S and the transformation−T is applied to the coordinates (x,y,z) to compute the coordinates(x,y,z) of the potential simplified vertex vs.

In an alternate embodiment of the present invention, tetrahedra u₀,u₁, .. . u_(m) may be identified in step 1203 as follows:

u₀=and the three vertices of t0

u₁=vs and the three vertices of t1

. . .

u_(m)=vs and the three vertices of tm

In this case, the equation of step 1203 would b e equated to zero instep 1205. This would lead to an alternate derivation of the sameequation established in step 1207. Step 1201 may be omitted.

Moreover, in an alternate embodiment of the present invention, in step1211, the coordinates x, y, and z of the potential simplified vertex vsare solved such that the coordinates x, y, z lie in the plane p_(s) andthe maximum distance from vs to each of the planes p₀,p₁ . . . p_(m) isa minimum.

A more detailed description of the operation of the system in performingthe secondary collapsibility tests of step 617 above is now set forthwith respect to FIG. 15. The secondary collapsibility tests preferablyinclude one or more geometric consistency tests and an error tolerancetest. As shown in FIG. 15, the second collapsibility tests preferablybegins operation in step 1501 with a first geometric consistency testperformed on the normals of the triangles of the edge star of theselected edge. More specifically, for each triangle in the edge star, ascalar product of the surface normal of the triangle prior tosimplification and the surface normal of the correspondingrepresentative triangle (which is generated in step 615 above) isdetermined. If any of such scalar products associated with the trianglesof the edge star is less than a predetermined maximum, the firstgeometric consistency test fails, the edge is marked as not collapsibleand the operation ends; otherwise operation continues to step 1503. Thepredetermined minimum scalar product may be specified by the user, andpreferably is greater than zero.

In step 1503, a second geometric consistency test is preferablyperformed that ensures that none of the resultant triangles aftersimplification exceed a predetermined criterion reflecting narrowness ofthe resultant triangles. More specifically, for each triangle in theedge star, a first coefficient Cp that reflects the narrowness of thetriangle is determined. And, for each triangle in the star of thepotential simplified vertex, a second coefficient Cs that reflects thenarrowness of the corresponding representative triangle is determined.Preferably, the coefficients Cp and Cs are determined as follows:$c\quad = \quad \frac{4\sqrt{3}a}{l_{0}^{2} + \quad l_{1}^{2} + l_{2}^{2}}$

where c is the coefficient Cp or Cs, a is the area of the triangle andl₀,l₁,l₂ are the lengths of the three sides of the triangle.

The minimum Cp value is then identified and multiplied by a minimumratio R. The result is then compared to the minimum Cs value. If theresult is greater than the minimum Cs value, the second geometricconsistency test fails; otherwise the second geometric consistency testpasses. The minimum ratio R is preferably greater than zero and may beset by user input. If the second geometric consistency test fails, theedge is marked as not collapsible and the operation ends; otherwiseoperation continues to step 1505.

In step 1505, an error tolerance test is performed wherein it isdetermined whether the simplified vertex generates an error bound thatexceeds a predetermined tolerance level. A more detailed description ofthe computation of such an error bound and the associated tolerancelevel is described below with respect to FIG. 16. If in step 1505 it isdetermined that simplified vertex generates an error bound that exceedsthe predetermined tolerance level, the edge is marked as not collapsibleand the operation ends. However, if in step 1505 the simplified vertexgenerates an error bound that does not exceed the predeterminedtolerance level, the edge is marked as collapsible and the operationends.

FIG. 16 illustrates the operations of the system in computing the errorbound and tolerance levels associated with a potential simplified vertexaccording to the present invention. More specifically, the operationbegins in step 1601 by reading the error value and tolerance value fromthe copies of the entries of the list of vertices 300 that correspond tothe vertices of the edge star of the selected edge. The error valueassociated with a vertex represents the upper bound estimate of an errorthat may arise as the result of the simplification operation. Thetolerance value associated with a vertex represents the maximum errorfor that vertex. The tolerance values assigned to the entries of thelist of vertices may be initialized manually by the user, or may beinitialized automatically. In addition, the error values associated withthe vertices may be initialized, for example, to a zero value. Thevertices of the edge star of the selected edge may be denoted as v0,v1 .. . vm. In this case, the error values corresponding to the vertices ofthe edge star of the selected edge may be denoted as ε₀, ε₁, . . .ε_(m), and the tolerance values corresponding to the vertices of theedge star of the selected edge may be denoted as τ_(o), τ₁, . . . τ_(m).

In step 1603, a tolerance value τ_(s) associated with the potentialsimplified vertex is determined. Preferably, the tolerance value τ_(s)is based upon the tolerance values of the vertices v0,v1 of thepotentially collapsible edge. For example, the tolerance value τ_(s) maybe determined as the minimum of the tolerance values τ_(o) and τ₁.

In step 1605, new error values δ₂,δ₃ . . . δ_(m) associated with thevertices of the edge star (excluding the vertices of the potentiallycollapsible edge v0,v1 of the selected edge and a new error value δ_(s)associated with the potential simplified vertex vs is computed. Threemethods have been developed to compute such error values. A moredetailed description of such methods is set for below.

In step 1607, one or more of the tolerance values τ_(s), τ₂, . . . τ_(m)and one or more of the new error values δ_(s), δ₂ . . . δ_(m) areevaluated to determine if the edge is collapsible. Preferably, the edgeis determined to be collapsible if the maximum of the new error valuesδ_(s),δ₂ . . . δ_(m) is less than the minimum of the tolerance valuesτ_(s), τ₂, . . . τ_(m). If the edge is determined not to be collapsible,the edge is marked as not collapsible and the operation ends. However,if the edge is determined to be collapsible, the edge is marked ascollapsible and the tolerance value τ_(s) of the potential simplifiedvertex computed in step 1603 is saved in the copy of the entry of thelist of vertices corresponding to the representative vertex for thepotential simplified vertex. As described above, one of the vertices v0of the selected edge is chosen as the representative vertex for thepotential simplified vertex. In addition, the new error valuesδ_(s),δ_(s) . . . δ_(m) computed in step 1605 are saved in thecorresponding copies of entries of the list of vertices. FIG. 17illustrates the assignment of error values and tolerance values to thevertices of the edge star prior to simplification. FIG. 18 illustratesthe assignment of error values and tolerance values to the vertices ofthe star of the potential simplified vertex that results fromsimplification.

In the first method, Method I, the new error values δ₂,δ₃ . . . δ_(m)that are associated with the vertices of the edge star that are not partof the potential collapsible edge (i.e., excluding v0 and v1) areequated to the old error values ε₂, ε₃, . . . . Thus δ₂=ε₂, δ₃=ε₃ . . .δ_(m)=ε_(m). In addition, the new error value δ_(s) associated with thepotential simplified vertex is given by the following formula:

δ_(s)=maximum (distance(vs,v0)+ε₀, distance(vs,v1)+ε₁).

The second and third methods, Method II and III, determine new errorvalues δ_(s),δ₂ . . . δ_(m) by first collecting a series of constraints,and by choosing new error values δ_(s),δ₂ . . . δ_(m) that satisfy theseconstraints. Generally, methods II and III generate the new error valuesδ_(s),δ₂ . . . δ_(m) as follows. First, an error volume is assigned tothe vertices of triangles of the edge star. Each triangle T_(i) of theedge star is partitioned into a plurality of planar polygons T_(ik). Foreach planar polygon T_(ik), an error volume is derived, which ispreferably computed by linear interpolation of the error volumesassigned to the vertices of the triangle T_(i). In addition, each of theplanar polygons T_(ik) is associated with one of the triangles, triangleQ_(k), of the star of the potential simplified vertex. Finally, an errorvolume for each triangle Q_(k) is derived that encloses the errorvolumes derived for the planar polygons T_(ik) associated with the giventriangle Q_(k). Preferably, the error volume derived for a giventriangle Q_(k) is computed by collecting constraints that depend uponthe error volumes of the associated planar polygons and the heightsbetween such planar polygons and the given triangle Q_(k). The errorvolumes are preferably composed of spheres whose centers are situated onthe corresponding triangles/planar polygons and whose radii correspondto the associated error values; however, the present invention is notlimited in this respect and may be utilized with any volumetricrepresentation. An example of using the present invention with aspherical error volumes is now set forth.

Methods II and III preferably begin by looping through each trianglet0,t1, . . . ,tm of the triangles of the edge star of the selected edge.For each triangle T_(i), the barycentric coordinates of the vertices vs,v2, v3, . . . ,vm with respect to the triangle T_(i) are computed. Adefinition of such barycentric coordinates is set forth in Hoffmann,“Geometric and Solid Modeling”, 1989, pp 51, herein incorporated byreference in their entirety. The barycentric coordinates of each vertexvs, v2, v3, . . . ,vm are denoted below as α_(s), α₂, . . . α_(m). Eachα_(j) is a vector of three coordinates denoted below as α_(j0), α_(j1),α_(j2). In addition, the heights (positive or negative) of vs, v2, . . .vm above the plane p_(i) of the triangle T_(i) are computed. The heightsare denoted below as h_(s), h₂, . . . h_(m). Recall that the normal ofplane p_(i) is denoted n_(i) and is stored in the entry of the list oftriangles 400 corresponding to the triangle T_(i).

The preferred method of determining the barycentric coordinates of avertex v_(j) with respect to a triangle T_(i) and of determining theheight h_(j) of the vertex v_(j) above the plane p_(i) is now set forth.As an example, consider the case when i=2 and j=5. Thus the triangleT_(i) corresponds to triangle T₂, α_(j) corresponds to α₅, and h_(j)corresponds to h₅. Other cases are treated similarly. As shown in FIG.20, triangle T₂ is composed of the vertices (v3, v0, v2), and its normalis n2. The normal n2 is directly available from the entry in the list oftriangles 400 corresponding to the triangle T₂. In this example, thevertex v5 may be represented as follows:

v5=α₅₀ v3+α₅₁ v0+α₅₂ v2+n2 h5  (1)

wherein α₅₀ is the barycentric coordinate referring to the first vertexof the triangle T₂, namely v3;

wherein α₅₁ is the barycentric coordinate referring to the secondvertex, v0;

wherein α₅₂ is the barycentric coordinate referring to the third vertex,v2; and

wherein v5,v3,v0 and n2 are vectors of three coordinates (x,y,z).

α₅₀, α₅₁, and α₅₂ are preferably determined using the followingprocedure. It is assumed that (v3,v0) is the shortest edge of thetriangle. If this is not true, (v3, v0, v2) are permuted circularlyuntil the first two vertices form the shortest edge, and thecorresponding backward permutation is stored. The vector v30 joining v3to v0, the vector v32 joining v3 and v2, and the vector v35 joining v3and v5 are computed. The following linear system of equations is solvedusing standard Normal Equations as described in Golub and Van Load,“Matrix Computations”, Johns Hopkins Press, 1989, pp 224-225,incorporated by reference above: ${\begin{bmatrix}{v30}_{x} & {v32}_{x} \\{v30}_{y} & {v32}_{y} \\{v30}_{z} & {v32}_{z}\end{bmatrix}\quad\begin{bmatrix}\alpha_{51} \\\alpha_{52}\end{bmatrix}}\quad = \quad \begin{bmatrix}{v35}_{x} \\{v35}_{y} \\{v35}_{z}\end{bmatrix}$

where the subscripts x, y and z naturally denote the x y and zcoordinates of the vectors v30, v32 and v35.

α₅₀, is set to (1−α₅₁α₅₂ ). If need be, the stored backward permutationis applied to the result to give the barycentric coordinates α₅₀, α₅₁,and α₅₂. Finally, the value of h5 is then determined by substituting thebarycentric coordinates α₅₀, α₅₁ and α₅₂ in equation (1) with the valuesdetermined above and solving for h5.

The barycentric coordinates of the vertices vs, v2, v3, . . . ,vm withrespect to the triangle T_(i) identify the positions of the projectionsof the vertices vs, v2, v3, . . . ,vm on the plane p_(i) on which thetriangle T_(i) lies. An operation is preferably performed that verifiesthat none of the edges the link of the potential simplified vertex haveprojected positions that intersect T_(i). A more detailed description ofthe determination of whether there is an intersection of the projectedposition of an edge v_(i) v_(j) and the triangle T_(i) is set forthbelow with reference to the derivation of type A constraints.

Then, for each triangle T_(i), one or more constraints are collected.There are preferably four classifications of constraints: Type A, TypeB, Type C and Type D. For a given triangle T_(i), there can be none orseveral type A constraints collected, there may be either one or twotype B constraints collected, there may be either zero or one Type Cconstraint collected, and there may be either zero or one Type Dconstraint collected. A detailed description of the constraintcollection operation is set forth below.

Method II consists of keeping the collected constraints (either A, B, Cand D) in a list, and by calling a linear programming solver thatminimizes the new error values δ_(s),δ₂ . . . δ_(m) subject to the listof constraints. The IBM Optimization Subroutine Library (OSL), which iscommercially available for a broad range of operatingsystems/programming environments, may be used as the linear programmingsolver.

Method III consists of equating the new error values δ₂,δ₃ . . . δ_(m)with the old error values ε₂, ε₃, . . . ε_(m). Thus, δ₂=ε₂, δ₃=ε₃ . . .δ_(m)=ε_(m). δ_(s) is computed as follows. Initially, the value 0 isassigned to δ_(s). Subsequently, each time a constraint of type A, B Cor D is encountered, a new potential value of δ_(s) is determined. Thevalue of δ_(s) is modified and set to the new potential value only ifthis new potential value is greater than the current value of δ_(s).

Type A Constraint:

A loop is performed on each edge defined by the vertex pairs of the starof the potential simplified vertex. The edges would include thefollowing: (vs, v2), (vs, v3), (vs, v4), . . . , (vs, vm). For eachedge, it is determined if there is a change of sign of any barycentriccoordinate between the two vertices of the edge. If there is a change inthe sign, the point vc wherein the barycentric coordinate is zero isdetermined. This point is called a “crossing”, because at this point theprojection of the edge intersects the triangle T_(i) from which thebarycentric coordinates were computed. For explanatory purposes,consider the case illustrated in FIG. 19 where i=2 and j=3. Thus, thetriangle T_(i) corresponds to triangle T₂, and α_(j) corresponds to α₃.Furthermore, suppose the first barycentric coordinate changes signbetween vs and v3. Since v3 is a vertex of the triangle T₂, the vectorof barycentric coordinates α₃ is (1,0,0), where α₃₀ is α_(s0)<0. It isknown that α_(s0) is negative because it is assumed that the firstbarycentric coordinate changes sign with α_(s0)*α₃₀<0. Next, a value λis defined as follows:$\lambda \quad = \quad {- \quad \frac{\alpha_{30}}{\alpha_{s0} - \alpha_{30}}}$

Newxt, the barycentric coordinates of vc, denoted α_(c), are computed asfollows:

α_(c0)=λ*α_(s0)+(1−λ)*α_(j0)

α_(c1)=λ*α_(s1)+(1−λ)*α_(j1)

α_(c2)=λ*α_(s2)+(1−λ)*α_(j2)

The next step is to determine if the three barycentric coordinates of vc(α_(c0), α_(c1), α_(c2)) are greater than or equal to zero, which meansthat vc projects on the boundary of the triangle T₂ (it can't projectinside since one barycentric coordinate is equal to zero by assumption).If this condition is satisfied, vc is indeed a crossing, and thecoordinates of vc are determined as follows:

vc ₀ =λ*vs ₀+(1−λ)*v3 ₀

vc ₁ =λ*vs ₁+(1−λ)*v3 ₁

vc ₂ =λ*vs ₂+(1−λ)*v3 ₂

Then, the height hc of vc above the plane p₂ is computed as follows:

hc=λ*hs+(1−λ)h3

The error ε_(c) at vc is then defined as:

ε_(c)=α_(c0)*ε₃+α_(c1)*ε₀+α_(c2)*ε₂

Note that in this equation, at least one of α_(c0), α_(c1), α_(c2) iszero, namely the first coordinate α_(c0). Then, a constraint on the newdelta values is determined as follows:

δ_(s)*λ+δ_(j)*(1−λ)≧|hc|+ε_(c)

where |hc| denotes the absolute value of hc.

Method II consists of adding this constraint to a list of constraintsinitially empty.

Method III consists of initializing the value of δ_(s) to zero andupdating the value of δ_(s) only if the new potential value of δ_(s)exceeds the current value of δ_(s). The new potential value of δ_(s) isdetermined as follows:$\delta_{s} = \quad \frac{\left( \left| {hc} \middle| \quad {{+ \quad \varepsilon_{c}} + \quad {\left( {\lambda \quad - \quad 1} \right)\delta_{j}}} \right. \right)}{\lambda}$

δj may be replaced with ε₁ in the expression, and the expressionbecomes:.$\delta_{s} = \quad \frac{\left( \left| {hc} \middle| \quad {{+ \quad \varepsilon_{c}} + \quad {\left( {\lambda \quad - \quad 1} \right)\varepsilon_{j}}} \right. \right)}{\lambda}$

Preferably, it is determined whether all edges for which a crossing wasidentified as described above can be circularly ordered without gaps. Ifthis condition is satisfied, then the process continues; otherwise, theedge is marked as not collapsible and the process terminates. Forexample, the following list of edges can be circularly ordered withoutgaps: (vs,v3), (vs,v4), (vs,v5); however, the following list cannot becircularly ordered without gaps: (vs,v3), (vs,v4), (vs,v6).

Type B Constraint:

The type B constraint is applied to both vertices v0 and v1. It isdetermined whether v0 or v1 or both v0 and v1 belong to the triangleT_(i). For explanatory purposes, consider the example discussed abovewith respect to FIG. 21 where v0 belongs to T₂. In a first step, theprojections of the triangles (vs,v2,v3), (vs,v3,v4) . . . onto thetriangle T₂ which contain the vertex v0 are identified. Such projectionsare denoted q2,q3, . . . qm below. It is possible that v0 projectsexactly onto one of the edges (vs, v2), . . . , (vs, v3). If this is thecase, vc is a crossing. However, it is preferably treated as Type Bconstraint.

The second step is to loop through the projections serially and locatethe first projection qk onto which v0 projects. Preferably, one of theedges of the first projection qk is an edge where a “crossing” isidentified as described above. If there is no projection onto which v0projects, then the configuration is “invalid” and the selected edge ismarked as not collapsible. However, if there is at least one projectiononto which v0 lies, a third step is performed.

For illustrative purposes, consider FIG. 21 where v0 lies inside theprojection of q3=(vs,v3,v4). The third step is to compute the closestpoint from v0 to the triangle (vs, v3, v4) as well as the vector ofbarycentric coordinates, denoted β, with respect to (vs, v3, v4).Preferably, the vector β is generated using the method described aboveto generate the vectors α_(i). In addition, the distance d0 from v0 tothe plane of (vs, v3, v4) is determined.

Finally, a fourth step is performed whereby a constraint is identifiedas follows:

δ_(s)*β₀+δ₃*β₁+δ₄*β₂≧d0+ε₀

where β₀, β₁, β₂ are the components of the vector β.

If need be, the constraint may be expressed as follows:$\delta_{s\quad} \geqq \quad \frac{{{- \delta_{3}}*\quad \beta_{1}} - \quad {\delta_{4}*\quad \beta_{2}} + \quad {d0}\quad + \quad \varepsilon_{0}}{\beta_{0}}$

The following operations may be used to determine if v0 projects onto aprojection qk (i.e., projection qk contains v0). First, the last twobarycentric coordinates of v0 are determined. The last two barycentriccoordinates may be either (0,0), (1,0) or (0,1) since v0 is one of thevertices of the triangle T₂, Then, the barycentric coordinates of thethree vertices of the projection qk under test, denoted below as(x0,y0), (x1,y1) and (x2,y2) is determined. An example of the vertices(x0,y0), (x1,y1) and (x2,y2) is shown in FIG. 23. For a given point withcoordinates (x,y), the given point is inside the projection qk if thearea of the following three triangles is greater than or equal to zero.The three triangles include:

1) the triangle (x0,y0), (x1,y1), (x,y)

2) the triangle (x1,y1), (x2,y2), (x,y)

3) the triangle (x2,y2), (x0,y0), (x,y)

The area of the triangle (x0,y0), (x1,y1), (x,y) may be represented bythe (½) the determinant of the following matrix (See O'Rourke“Computational Geometry in C” Cambridge University Press, 1994pp.30−31): $\begin{bmatrix}x_{0} & x_{1} & x \\y_{0} & y_{1} & y \\1 & 1 & 1\end{bmatrix}$

Similar steps may be performed for the other two triangles. This leadsto the following three equations:

x(y0−y1)+y(x1−x0)+x0y1−y0x1≧0

x(y1−y2)+y(x2−x1)+x1y2−y1x2≧0

x(y2−y0)+y(x0−x2)+x2y0−y2x0≧0

If the three values obtained by replacing x and y with the actualbarycentric coordinates of v0 (which are either (0,0), (1,0) or (0,1))are positive, then it is determined that v0 is inside the projection qk.Otherwise, it is determined that v0 is outside the projection qk.

Type C Constraint:

The type C constraint verifies whether the potential simplified vertexvs projects inside the triangle T_(i). This constraint is holds true ifall three barycentric coordinates of the potential simplified vertex vswith respect to the vertices of T_(i) are greater than or equal to zero.Preferably, the barycentric coordinates (α_(s0), α_(s1), α_(s2)) ) ofthe vertices of the triangle T_(i) are generated using the methoddescribed above to generate the vectors α_(i). For example, consider thetriangle T4 whose vertices comprise (v1,v0,v5) as shown in FIG. 22. Inthis case, if this test is true, then the constraint is generated asfollows:

δ_(s) ≧|h _(s)|+α_(s0)*ε₁+α_(s1)*ε₀+α_(s2)*ε₅

where h_(s) is the height of potential simplified vertex vs above theplane of p4.

Type D Constraint:

The type D constraint is activated after all triangles have beenprocessed, exclusively in the case when no type C constraint wasgenerated. In that case, an edge, denoted f, is selected from all edgesbelonging to the triangles of the edge star less the edges of the linkof the edge star. The edge f is the closest edge to the potentialsimplified vertex vs. For the edge f, the potential simplified vertex vsis projected onto f, and the distance df between vs and its projectiononto f, denoted vf, is computed. The barycentric coordinates of vf,denoted α_(k) and α₁, with respect to the vertices of f (denoted vk andv1) are computed. The vertices vk and v1 are associated with errorvalues ε_(k) and ε₁. A constraint is then generated as follows:

δ_(s) ≧dƒ+α _(l)*ε_(l)+α_(k)*ε_(k)

In an alternate embodiment of the present invention, the followingmethods can be used to collect constraints on the new error values,δ_(i). The processing of such constraints follows either Method II orMethod III as set forth above. The difference between Method IV andMethod V is that Method IV processes constraints according to Method IIand Method V processes constraints according to Method V.

As before, a mapping is defined between the triangles of the edgeconfiguration before simplification, represented by the edge star(denoted below as e0*), and the triangles of the configuration aftersimplification, represented by the star of the simplified vertex(denoted below as vs*), as illustrated in the FIG. 24. However, thismapping is not obtained by looping on each triangle and projecting theconfiguration separately on each triangle as accomplished in Methods IIand III.

More specifically, Methods IV and V construct a tiling of the star e0*that is in one to one mapping with a tiling of the star vs*. The mappingis one to one because the two tilings have corresponding vertices andcorresponding edges. The following description indicates howcorresponding vertices are identified, which is sufficient forimplementing this method. In the beginning of the process, the initialtiling of e0* contains the vertices of e0* and the edges of e0*. Thesame applies to the initial tiling of vs*. The tiling begins byassigning three arbitrary vertex-triangle correspondences. The tiling isthen completed by determining corresponding “edge crossing points” whencomputing the shortest distance between edges. Edge crossing points areprecisely defined below.

First, determine the closest point to the vertex vs on the triangles t0and t1. The position of this point is obtained by projecting vs on botht0 and t1 as set forth above and by retaining the projected vertex thatis the closest to vs. This point is denoted ws. The couple of points(vs, ws) is associated with the distance hs from vs to ws as well as thebarycentric coordinates of ws with respect to the vertices of thetriangle of which ws lies. These barycentric coordinates are denotedalpha_(—)0, alpha_(—)1 and alpha_(—)2. For instance, in the case of FIG.24, ws can be expressed as follows:

ws=alpha_(—)0 v0+alpha_(—)1 v1+alpha_(—)2 v2,

whereby ws, v0, v1 and v2 denote the vectors representing the 3Dcoordinates of each vertex.

The constraint associated with (vs,ws) is derived as follows:

delta_s=hs+alpha_(—) x20 epsilon_(—)0+alpha_(—)1 epsilon_(—)1+alpha_(—)2epsilon_(—)2.

The vertex ws as well as the three edges (ws,v0) (ws,v1) and (ws, v2)are added to the tiling of e0*.

Then determine the closest point to the vertex v0 on the triangles q2 .. .qr−1. The position of this point is obtained by projecting v0 on eachtriangle of the sequence q2 . . . qr−1 and by retaining the projectedvertex that is closest to vs. This is denoted point w0. The couple ofpoints (v0,w0) is associated with epsilon0, the distance h0 between v0and w0 and the barycentric coordinates denoted alpha of w0 with respectto the vertices of the triangle on which w0 lies. For instance, in theFIG. 24, these are vertices vs, v3 and v4. The constraint associatedwith (v0,w0) is derived as:

alpha_(—)0 vs+alpha_(—)1 v3+alpha2 v4=h0+epsilon0

The vertex w0 as well as the three edges (w0,vs) (w0,v3) and (w0, v4)are added to the tiling of vs*.

Then determine the closest point to the vertex v1 on the triangles qr .. . qm. The position of this point is obtained by projecting v0 on eachtriangle of the sequence qr . . . qm and by retaining the projectedvertex that is closest to vs. This is denoted point w1. The couple ofpoints (v1,w1) is associated with epsilon1, the distance h1 between v1and w1 and the barycentric coordinates denoted alpha of w1 with respectto the vertices of the triangle on which w1 lies. For instance, in theFIG. 24, these are vertices vs, vm−2 and vm−1. The constraint associatedwith (v1,w1) is derived as:

alpha_(—)0 vs+alpha_(—)1 vm−2+alpha2 vm−1=h1+epsilon1.

The vertex w1 as well as the three edges (w1,vs) (w1,vm−2) and (w1,vm−1) are added to the tiling of vs*.

Then the vertex opposite v0 and v1 inside the triangle t0 is denoted vr.v2 and vr define the beginning and end of two sequences in the verticesof the link of e0: namely v2,v3 . . . ,vr and vr, vr+1, . . . . ,vm,v2,which are denoted by the lower and upper sequences of vertices. Thisnotation is illustrated in the FIG. 24. For each edge linking vs and anyvertex of the lower sequence of vertices (for instance the edge (vs,v3)), each edge belonging to e0* that is adjacent to v0 (for instance,the edge (v0, v2)) is visited, and the shortest distance to the edges isdetermined. The shortest distance occurs for a point vc belonging to(vs,v3) and a point wc belonging to (v2,v0). If either vc or wc is notan endpoint of its corresponding edge, such pair of points (vc,wc) iscalled an edge crossing. For each such edge crossing, a constraint isderived by first computing the values hc, lambda_c and mu_c as follows:

hc=distance (vc,wc),

vc=lambda_c vs+(1-lambda_c) v3,

wc=mu_c v2 +(1-mu_c)v0,

and by then deriving the following constraint equation:

lambda_c delta_s+(1-lambda_c)delta_(—)3=hc+mu_c epsilon_(—)2+(1-mu_c)epsilon_(—)0

For each such edge crossing point identified, the edge (v2,v0) isremoved from the tiling of e0*, and the vertex wc is added. In addition,the four edges (v2,wc) and (wc,v0), (ws,wc) and (wc,v3) are added. Also,the edge (vs,v3) is removed from the tiling of vs*, and the vertex vcand the four edges (vs,vc) and (vc,v3), (v2,vc) and (vc,w0) are added.

This sequence of operations is repeated for each edge linking vs and theupper sequence of vertices by visiting in turn each edge belonging toe0* that is adjacent to v1.

In an alternate embodiment of the present invention, one or more datavalues are associated with each vertex of the original surface. Forexample, he data values associated with a given vertex may represent thecolor of that vertex or a texture coordinate, such as a (u,v) coordinatefor that vertex. According to the present invention, such data valuesare linearly interpolated between vertices, and the difference betweenthe data associated with the original surface and the data associatedwith the simplified surface will be bounded by a maximum value gamma.The value gamma may be specified by the user. The data values areassumed to be real numbers; however, the present invention may be usedwith complex numbers or vectors provided that each coordinate is treatedas a separated data.

First, the data values c0, c1, . . . cm are associated with the verticesof an edge star. For instance, v0,v1 , . . . vm are associated with thedata values c0,c1, . .. ,cm before simplification. There is a data errorvolume associated with the data. At each vertex a positive data errorvalue phi_i is specified. It indicates that the true data value at thatvertex is somewhere between ci−phi_i and ci+phi_i.

New data values are determined at the vertices after simplification: vs,v2, . . . , vm, denoted d_s, d_2 . . . , vm, as well as new data errorvalues denoted psi_s, psi_2, . . . , psi_m.

The principle underlying the method is very similar with what was setforth above. Constraints are collected on the d_i values and the maximumdifference between linearly interpolated d data values and linearlyinterpolated c data values is minimized. Similarly to Method III above,the data values as well as the data error values at the link of the edgestar under test are copied:

d₁₃2=c_(—)2, d_(—)3=c_(—)3, . . . , d_m=c_ml

psi_(—)2=phi_(—)2, psi_(—)3=phi_(—)3, . . . , psi_m=phi_m.

The new potential data value d_s and the new potential data error valueare then determined as follows. The constraints used may be type A, typeB, type C, or type D constraints as described above, or similarconstraints. The type A, type B, type C, or type D constraints would berewritten using the following three rules.

1) Each h_i value that appears in any of the A,B,C,D, constraint isreplaced with 0;

2) Each epsilonj value is replaced with c_j+phi_j;

3) Each delta_k value is replaced with d_k+psi_k.

The sum (d_k+psi_k) is isolated from the constraints as was done abovefor Method III. Each such obtained constraint is numbered using acounter n, and for each n, we determine lamda_n and w_n such that theconstraint n can be rewritten as the following equation.

lambda_n(d_s+psi_s)=w_n

The following expression is minimized: MAX |lambda_n*x−w_n . Thisproduces a value x, which is assigned to d_s. The maximum is reached fora given n, denoted n_min.

The following expression is then evaluated to give psi_s:

psi_s=|lambda_n_min*d_s−w_n|/lambda_n_min

If psi_s is larger than gamma, the edge is marked as not collapsible;otherwise the edge is marked as collapsible.

In an alternate embodiment of the present invention, the boundary testof step 1101 of FIG. 11 may be modified such that boundary edges can besimplified. Preferably, such simplification preserves the length of thesurface boundary after simplification. A boundary edge is such that bothvertices of the edge are boundary vertices. Boundary vertices weredefined precisely above.

FIG. 25 shows e0=(v0,v1) which is a boundary edge under test. vr is thevertex located to the right of v0 on the boundary. v1 is the vertexlocated to the left of v1 on the boundary.

First, a valence test similarly to step 1103 is performed. The valencetest for this particular case is:

2<=v(v0)+v(v1)−2<=valence_max

Then, an ellipsoid is determined whose focal points are vr and v1 suchthat any point m on the ellipse will satisfy the property relating thefollowing distances:

dist(vr,m)+dist(v1,m)=dist(vr,v0)+dist(v0,v1)+dist(v1,v1)=2a

Such ellipsoid can be obtained using standard methods. On suchellipsoid, it is determined the location of the point that minimizes thesum of distances to the three lines determined by (vr,v0), (v0,v1) and(v1,v1), or that minimizes the sum of distances to a set of points orlines that are in the vicinity of vr,v0,v1 and v1. This point is denotedvm.

The topological consistency and geometric consistency tests of steps1107 and 1503 are then performed. The error tolerance test of step 1607as set forth in Method IV or V above is then performed, ignoring allsteps referring to the lower sequence of vertices and the lower sequenceof triangles.

If all the previous tests are passed successfully, the boundary edge ismarked collapsible and the edge collapse operation is performed usingthe representative updates as indicated by arrows in the FIG. 25. If anyof the tests fails, the boundary edge is marked as not collapsible andthe process ends.

In summary, the simplification technique of the present inventionprovides three key advantages.

First, the approximation of the original surface preserves a solidvolume. More specifically, if the original surface has a boundary, thenthe volume enclosed by a composite closed surface is preserved. Thecomposite closed surface is preferably generated by juxtaposing to afirst surface an arbitrary surface that shares the same boundary. Thisprovides advantages by limiting the shrinkage of the original surface.Shrinkage degrades the visual appearance of the surface for graphicalapplications, and may degrade the accuracy of the surface such that themodel becomes useless, for example, in medical imaging applications.

Second, the simplification technique of the present invention ensuresthat, at each point of the original surface, a first distance to theclosest point on the simplified surface will be less than the tolerancespecified for that point. Moreover, at each point of the simplifiedsurface, a second distance to the closest point on the original surfacewill be less than the minimum of the tolerances at any of thesurrounding vertices. This implies that the second distance will be lessthan the maximum tolerance that was specified for any point on thesurface. An upper bound to the second distance at each point of thesimplified surface is preferably represented by a positive error valueat that point. Such error values are preferably derived from the widthof an error volume associated with the simplified surface. Such errorvolume is preferably obtained by sweeping a sphere of linearly varyingradius over the simplified surface. The technique presented isadvantageous because such error values are point specific and may bere-used for distance and intersection computations, and also becausesuch error volumes may be re-used for subsequent simplifications.

Third, the simplification technique of the present invention favors thecreation of near equilateral triangles over narrow triangles. This isadvantageous because near equilateral triangles produce better visualresults when rendering and allow more accurate numerical computationsfor finite element applications or other numerical techniques.

The overall computational complexity of the method is subquadratic inthe number of edges of the surface, making it affordable for largesurface models. Processing of models containing as many as 500 Ktriangles or more takes a few minutes on a UNIX workstation.

I claim:
 1. In a computer system wherein objects are represented by triangles defined by coordinates of vertices, a method for generating coordinates of a simplified vertex based upon coordinates of vertices adjacent to a first vertex and to a second vertex that define an edge of said triangles, the method comprising the steps of: identifying a set of triangles that are adjacent to said edge; generating a set of tetrahedra corresponding to said set of triangles, wherein each tetrahedron in the set of tetrahedra is defined by three vertices of a corresponding triangle and a common point shared by all tetrahedra in the set of tetrahedra; calculating a first volume associated with said set of tetrahedra; and determining coordinates of said simplified vertex such that a second volume associated with said simplified vertex corresponds to said first volume.
 2. The method of claim 1, wherein said set of triangles is an edge star of said edge. triangles.
 3. The method of claim 2, wherein said common point is a centroid of said set of triangles.
 4. The method of claim 2, wherein said second volume comprises a volume of a second set of tetrahedra.
 5. The method of claim 4, wherein each tetrahedron of said second set of tetrahedra is defined by said simplified vertex, said common point and two vertices of said triangles of said set of triangles.
 6. The method of claim 1, wherein coordinates of said simplified vertex are located such the said first volume is equal to said second volume.
 7. The method of claim 1, wherein coordinates of said simplified vertex are based upon coordinates of planes associated with said set of triangles.
 8. The method of claim 7, wherein said coordinates of said simplified vertex are located such that squared distances from said simplified vertex to said planes is a minimum.
 9. A program storage device for use in a computer system wherein objects are represented by triangles defined by coordinates of vertices, the program storage device readable by a machine, tangibly embodying a program of instructions executable by the machine to perform method steps for generating coordinates of a simplified vertex based upon coordinates of vertices adjacent to a first vertex and to a second vertex that define an edge of said triangles, the method steps comprising: identifying a set of triangles that are adjacent to said edge; generating a set of tetrahedra corresponding to said set of triangles, wherein each tetrahedron in the set of tetrahedra is defined by three vertices of a corresponding triangle and a common point shared by all tetrahedra in the set of tetrahedra; calculating a first volume associated with said set of tetrahedra; and determining coordinates of said simplified vertex such that a second volume associated with said simplified vertex corresponds to said first volume.
 10. The program storage device of claim 9, wherein said set of triangles is an edge star of said edge.
 11. The program storage device of claim 10, wherein said common point is a centroid of said set of triangles.
 12. The program storage device of claim 11, wherein said second volume comprises a volume of a second set of tetrahedra.
 13. The program storage device of claim 12, wherein each tetrahedron of said second set of tetrahedra is defined by said simplified vertex, said common point and two vertices of said triangles of said set of triangles.
 14. The program storage device of claim 9, wherein coordinates of said simplified vertex are located such the said first volume is equal to said second volume.
 15. The program storage device of claim 9, wherein coordinates of said simplified vertex are based upon coordinates of planes associated with said set of triangles.
 16. The program storage device of claim 15, wherein said coordinates of said simplified vertex are located such that squared distances from said simplified vertex to said planes is a minimum.
 17. The method of claim 1, wherein the first volume is calculated by summing the volume contribution for each tetrahedron in the set of tetrahedra.
 18. The method of claim 17, wherein the volume contribution for a given tetrahedron is determined by: forming a matrix based upon a translation to three vertices of the given tetrahedron; computing a determinant of the matrix; and applying a scale factor to the determinant.
 19. The program storage device of claim 9, wherein the first volume is calculated by summing the volume contribution for each tetrahedron in the set of tetrahedra.
 20. The program storage device of claim 19, wherein the volume contribution for a given tetrahedron is determined by: forming a matrix based upon a translation to three vertices of the given tetrahedron; computing a determinant of the matrix; and applying a scale factor to the determinant. 